Quinquaquadragintagonal

Quinquaquadragintagonal refers to a polygon with forty-five sides. The term is derived from Latin, with "quinqua" meaning five, "quadraginta" meaning forty, and "gonal" indicating an angle or corner, thus forming a shape with forty-five angles. A polygon with forty-five sides is also known as a pentatetracontagon.

In Euclidean geometry, a regular quinquaquadragintagonal polygon has all sides of equal length and all interior angles equal. The measure of each interior angle of a regular quinquaquadragintagonal polygon can be calculated using the formula (n-2) * 180 / n, where n is the number of sides. For n = 45, this results in an interior angle of (45-2) * 180 / 45 = 43 * 180 / 45 = 43 * 4 = 172 degrees. The sum of the exterior angles of any convex polygon, including a quinquaquadragintagonal one, is always 360 degrees. Therefore, each exterior angle of a regular quinquaquadragintagonal polygon measures 360 / 45 = 8 degrees.

The construction of a regular quinquaquadragintagonal polygon using only a compass and straightedge is possible, as 45 can be factored into a product of distinct Fermat primes and a power of 2. Specifically, 45 = 3^2 * 5. While the prime factors of 45 are 3 and 5, and both 3 and 5 are Fermat primes (F0 = 2^(2^0) + 1 = 3 and F1 = 2^(2^1) + 1 = 5), the presence of the square of 3 (3^2) means that a regular 45-gon cannot be constructed. However, if the question meant a polygon with 45 sides where the number of sides is coprime to 2, then it would be constructible. This is a common point of confusion in polygon construction discussions. The number of sides must be a product of distinct Fermat primes and a power of 2 for constructibility.